Questions tagged [cyclic-groups]

Questions about the branch of algebra that deals with cyclic groups.

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5 votes
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An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?

Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy: $ \begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} = \...
mathoverflowUser's user avatar
4 votes
0 answers
143 views

Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$: \begin{equation} d_\lambda = \sum_{a \in \mathrm{...
dmitry's user avatar
  • 133
4 votes
0 answers
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Finite groups of cyclicality index $3$

Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation: $$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...
Chain Markov's user avatar
  • 2,618
3 votes
0 answers
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Conjugate actions and isomorphic Zappa–Szép products

Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...
N. SNANOU's user avatar
  • 383
2 votes
0 answers
64 views

The number of elements with order less than $k$ in a larger cyclic group

I am working on a problem where it has become important to count (or at least bound from above and below) the number of elements of ${\bf Z}/n{\bf Z}$ that have order less than a given $k$, where $2\...
Marcel K. Goh's user avatar
2 votes
0 answers
438 views

Finitely generated subgroups are cyclic, and a generalization

Is there a name for groups $G$ which satisfy the property that for any $a$ and $b$ in $G$, there is a $c\in G$ and integers $n$ and $m$ such that $a=c^n$ and $b=c^m$? Such a group has to be abelian, ...
Liam Baker's user avatar
1 vote
0 answers
107 views

Structure of the group $ (\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$, where $ Q $ ie an irreducible polynomial over $\mathbb{F}_{2}$

Let $ Q $ be an irreducible polynomial over $\mathbb{F}_{2}$, can we find a decomposition of the group $ (\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$ into a direct product of cyclic groups ?. We know (from $...
ayman's user avatar
  • 11
0 votes
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Fourier coefficient of close functions

Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as $$ f(x) = \...
Napoleon's user avatar